Remarks on Pell's equation

[Larry Hammick]

Given a positive integer and a square matrix of this form
a Db
b a
and another matrix of the same form
x Dy
y x
the product of those two has the same form, voila
ax+Dby .. D(bx+ay)
bx+ay .. ax+Dby
Also, the product of the two given determinants is the determinant of
the product of the two given matrices. Leaving some details to the
reader, we thus get (by transport of structure) a group structure on
the set of lattice points on the parabola
(1) xx - Dyy = 1
or (same thing) a group structure on the set of solutions of Pell's
equation.

The traditional account of Pell's equation uses the ring of real
numbers of the form u+sqrt(D)v where u and v are integers, but the
matrices avoid these unsightly square roots and, at the same time,
they impart an attractive geometric tone to the subject.

Consider now the set of solutions of (1) having x>0. If D is
squarefree it contains more than just the identity element (1,0) of
the aforementioned group. The set of solution-matrices is generated by
the single element for which y>0 and x is as small as possible, and if
we fix D=2 that generator is
3 4
2 3
The first few powers of this matrix give the first few solutions (x,y)
of (1), namely
(1,0) (3,2) (17,12) (99,70) (577,408)
Notice that the these triangles:
(0,0) (1,0) (3,2)
(0,0) (3,2) (17,12)
(0,0) (17,12) (99,70)
(0,0) (99,70) (577,408)
etc.
all have the same area, 1.

And I think I'll leave it at that, for anybody who wants to pursue
this little recreational topic.

[Larry Hammick]

For more on the topic of group laws on conics, see the papers under
"Conics" at Franz Lemmermeyer's site:

http://www.fen.bilkent.edu.tr/~franz/publ.html

Hat tip: Bill Dubuque

[Gerry]

On Aug 22, 12:57 pm, Larry Hammick <larryhamm...@telus.net> wrote:
> Given a positive integer and a square matrix of this form
> a Db
> b a
> and another matrix of the same form
> x Dy
> y x
> the product of those two has the same form, voila
> ax+Dby .. D(bx+ay)
> bx+ay   .. ax+Dby
> Also, the product of the two given determinants is the determinant of
> the product of the two given matrices. Leaving some details to the
> reader, we thus get (by transport of structure) a group structure on
> the set of lattice points on the parabola
> (1) xx - Dyy = 1
> or (same thing) a group structure on the set of solutions of Pell's
> equation.

ITYM hyperbola.
--
GM

[Larry Hammick]

> > ... the set of lattice points on the parabola

> > (1) xx - Dyy = 1
> > or (same thing) a group structure on the set of solutions of Pell's
> > equation.
>
> ITYM hyperbola.
> --
> GM

Oops, yes.